Skip to main content

Full text of "Men Of Mathematics"

See other formats

transformations of indefinite ternary quadratic forms were
identical with those of non-Euclidean geometry.
On returning to Caen, I reflected on this result and
deduced its consequences; the example of quadratic forms
showed me that there were Fuchsian groups other than
those corresponding to the hypergeometric series; I saw
that I could apply to them the theory of thetafuchsian
functions, and hence that there existed thetafuchsian
functions other than those derived from the hypergeo-
metric series, the only ones I had known up till then.
Naturally I set myself the task of constructing all these
functions, I conducted a systematic siege and, one after
another, carried all the outworks; there was however one
which still held out and whose fall would bring about that
of the whole position. But all my efforts served only to
make me better acquainted with the difficulty, which in
itself was something. All this work was perfectly conscious.
At this point I left for Mont-Valerien, where I was to
discharge my military service. I had therefore very
different preoccupations. One day, while crossing the
boulevard, the solution of the difficulty which had stopped
me appeared to me all of a sudden. I did not seek to go into
it immediately, and it was only after my service that I
resumed the question. I had all the elements, and had only
to assemble and order them. So I wrote out my definitive
memoir at one stroke and with no difficulty.
Many other examples of this sort of thing could be given from
his own work, he says, and from that of other mathematicians
as reported in ISEnseignement Mathtmatique. From his experi-
ences he believes that this semblance of 'sudden illumioatian
[is] a manifest sign of previous long subconscious work9, and he
proceeds to elaborate his theory of the subconscious mind and
its part in mathematical creation. Conscious work is necessary
as a sort of trigger to fire off the accumulated dynamite wbieb
the subconscious has been excreting - he does not put it so, bat
what he says amounts to the same. But what is gained in the
way of rational explanation if, following Poincare*, we foist off
on the 'subconscious mind*, or the 'subliminal self, the very
activities which it is our object to understand? Instead of eo-
dowing this mysterious agent with a hypothetical tact enabling